Generating jigsaw pieces from svg curves
Knife project curves

The main problem with your object is that the pieces are not defined by a boundary. The curves are not intersecting each other: they are just drawn lines, and expecially in the "crossroads", there is't a common point between them.

 

In order to get the intersection, I would suggest to take advantage of the Knife project tool to a plane object (Shift+A Add mesh: Plane) of the right size. It has some degree of approximation but should be good enough for the provided example. Remember to keep the whole mesh inside the 3D view and run the command in ortho top view as it is view dipendent.

You don't need to convert your objects to meshes or join them: just select them all and run the command in edit mode (it might take a but of time to complete).



Finish by:


calling  the Edges Split operator (you can find it in the Crtl+E menu) in order to disjoin the faces
and then finally run the Separate by Loose parts (P) command on the remaining faces.